This brief summary captures the main operational features
of the traditional `optimal state-dependent cloning' and highlights the main issue with this
approach: it leaves no room to leverage operational equivalences to further study its potential non-classical
aspects. To fix that, we follow Ref. [1]} and exploit another operational consequence of the purity of \(\mathinner {|{a}\rangle }\) , \(\mathinner {|{b}\rangle }\) : the existence of preparations \(P_{a^\perp }\) , \(P_{b^\perp }\)  satisfying \(p(M_a|P_{a^\perp }) = p(M_b|P_{b^\perp }) = 0\)  and such that the mixture \( P_a/2 + P_{a^\perp }/2\)  (tossing a fair coin and following either \(P_a\)  or \(P_{a^\perp }\) ) is operationally equivalent to the mixture \( P_{b}/2 + P_{b^{\perp }}/2\) : \(P_a/2 + P_{a^\perp }/2 \simeq P_{b}/2 + P_{b^{\perp }}/2\) . In the idealized quantum experiment one observes this operational statistics by preparing pure states \(\mathinner {|{a^\perp }\rangle }\) , \(\mathinner {|{b^\perp }\rangle }\)  in the span of {\(\mathinner {|{a}\rangle }\) , \(\mathinner {|{b}\rangle }\) } and satisfying \(\!\left\langle a \vphantom{a^\perp }| a^\perp  \vphantom{a}\right\rangle \! = \!\left\langle b \vphantom{b^\perp }| b^\perp  \vphantom{b}\right\rangle \!=0\)  as well as \(\frac{1}{2} \left|a\right\rangle \!\left\langle a\right| + \frac{1}{2} \left|a^\perp \right\rangle \!\left\langle a^\perp \right| = \frac{1}{2} \left|b\right\rangle \!\left\langle b\right| + \frac{1}{2} \left|b^\perp \right\rangle \!\left\langle b^\perp \right|\) . The same discussion can be repeated for each of the pairs \(\lbrace (a,b), (\alpha , aa), (\beta , bb) \rbrace \) .
